Rotation Numbers, Boundary Forces and Gap labelling

We review the Johnson-Moser rotation number and the $K_0$-theoretical gap labelling of Bellissard for one-dimensional Schr\"odinger operators. We compare them with two further gap-labels, one being related to the motion of Dirichlet eigenvalues, the other being a $K_1$-theoretical gap label. We argue that the latter provides a natural generalisation of the Johnson-Moser rotation number to higher dimensions.Comment: 10 pages, version accepted for publicatio

On stronger conjectures that imply the Erd\"os-Moser conjecture

The Erd\"os-Moser conjecture states that the Diophantine equation $S_k(m) = m^k$, where $S_k(m)=1^k+2^k+...+(m-1)^k$, has no solution for positive integers $k$ and $m$ with $k \geq 2$. We show that stronger conjectures about consecutive values of the function $S_k$, that seem to be more naturally, imply the Erd\"os-Moser conjecture.Comment: 7 page

Bringing Sexy Back: Unauthorized Film Editing, Copyright, and How Removing Reproductive Acts Violates Reproduction Rights

The Multicolored Asian Ladybird, Harmonia axyridis, is an extremely successful invasive species. Here we suggest that, in addition to many other traits, the dorsal spines of its larvae contribute to their success, as suggested by behavioral observations of agonistic interactions between H. axyridis and European coccinellids. In coccinellids, the role of dorsal spines in these interactions has been poorly studied and they could be a physical protection against intraguild predators. Dorsal spines of second instar H. axyridis larvae were removed with micro-scissors, which resulted in spineless larvae after molting (spineless group). These larvae were then exposed to starved Coccinella septempunctata larvae. Two control categories were also submitted to interactions: H. axyridis larvae with all their spines (control group) and with their spines, but injured by pin stings (injured group). Spine removal at the second instar did not hamper H. axyridis development. The bite rate by C. septempunctata was significantly higher on the spineless H. axyridis and more dorsally located compared to the control and injured groups, while no bite rate difference was observed between the injured and the control group. Our results suggest that in addition to behavioral and chemical defenses, the dorsal spines play a significant protective role against bites. Therefore, spines in ladybirds could be considered as a morphological defense against intraguild predation. In H. axyridis, these defenses might contribute to its success in food resources already exploited by other guild members and thus further facilitate the invasion of new areas.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Formal Solutions of a Class of Pfaffian Systems in Two Variables

In this paper, we present an algorithm which computes a fundamental matrix of formal solutions of completely integrable Pfaffian systems with normal crossings in two variables, based on (Barkatou, 1997). A first step was set in (Barkatou-LeRoux, 2006) where the problem of rank reduction was tackled via the approach of (Levelt, 1991). We give instead a Moser-based approach. And, as a complementary step, we associate to our problem a system of ordinary linear singular differential equations from which the formal invariants can be efficiently derived via the package ISOLDE, implemented in the computer algebra system Maple.Comment: Keywords: Linear systems of partial differential equations, Pfaffian systems, Formal solutions, Moser-based reduction, Hukuhara- Turritin normal for